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3.6. Definition. The Maslov index µ Θ (x) ∈�of x = (x1, x2) ∈ P Θ (H1 ⊕ H2) is the integer
µ Θ (x) := µ(N ∗ ∆ ΘÊn, graphCΦ) − n
2 .
Since the intersection of the Lagrangian subspaces N ∗∆ΘÊn and graphCΦ(0) = graphC has
dimension 3n, the relative Maslov index µ(N ∗∆ΘÊn, graphCΦ) differs from 3n/2 by an integer (see
[RS93], Corollary 4.12), so µ Θ (x) is an integer. The fact that this definition does not depend
on the choice of the trivialization is proved in [APS08], in the more general setting of arbitrary
non-local conormal boundary conditions.
By (30), the elements of PΘ (H1 ⊕ H2) are critical points of the action functional�H1⊕H2 =
�H1
⊕�H2 on the space of pairs of curves x1, x2 : [0, 1] → T ∗M whose four end points have the same
projection on M. We endow M × M with the product metric, and T ∗M2 with the corresponding
almost complex structure, still denoted by J. Let x− = (x − 1 , x−2 ) and x+ = (x + 1 , x+ 2 ) be two
elements of PΘ (H1 ⊕ H2), and let M Θ ∂ (x− , x + ) be the space of maps u ∈ C∞ (Ê×[0, 1], T ∗M2 )
which solve the Floer equation
∂J,H1⊕H2(u) = 0,
together with the boundary and asymptotic conditions
The following result is proved in section 5.10.
(u(s, 0), −u(s, 1)) ∈ N ∗ ∆ Θ M, ∀s ∈Ê,
lim
s→±∞ uj(s, ·) = x ± j , ∀j = 1, 2.
3.7. Proposition. For a generic choice of H1 and H2 the set M Θ ∂ (x− , x + ) is a smooth manifold
of dimension µ Θ (x − ) − µ Θ (x + ). These manifolds can be oriented in a coherent way.
Let us deal with compactness issues. Lemma 3.1 together with assumption (H0) Θ implies that
for every A ∈Êthe set of (x1, x2) ∈ P Θ (H1 ⊕ H2) with�H1(x1) +�H2(x2) ≤ A is finite.
Assumptions (H1) and (H2) allow to prove the following compactness result for solutions of
the Floer equation (see section 6.1).
3.8. Proposition. Assume that H1 and H2 satisfy (H1), (H2). Then for every x − , x + ∈
P Θ (H1 ⊕ H2), the space M Θ ∂ (x− , x + ) is pre-compact in C ∞ loc (Ê×[0, 1], T ∗ M 2 ).
If we now assume that H1 and H2 satisfy (H0) Θ , (H1), and (H2), we can define the Floer
complex in the usual way. Indeed, for x− , x + ∈ PΘ (H1 ⊕H2) such that µ Θ (x− )−µ Θ (x + ) = 1, we
define nΘ ∂ (x− , x + ) ∈�to be the algebraic sum of the orientation sign associated to the elements
of M Θ ∂ (x− .x + ), and we consider the boundary operator
∂ : F Θ k (H1 ⊕ H2) → F Θ k−1(H1 ⊕ H2), x − ↦→ �
n Θ ∂ (x − , x + )x + ,
x + ∈P Θ (H1⊕H2)
µ Θ (x + )=k−1
where F Θ k (H1 ⊕ H2) denotes the free Abelian group generated by the elements x ∈ P Θ (H1 ⊕ H2)
with µ Θ (x) = k.
The resulting chain complex F Θ (H1 ⊕H2, J) is the Floer complex associated to figure-8 loops.
If we change the metric on M - hence the almost complex structure J on T ∗ M 2 - and the orientation
data, the Floer complex F Θ (H1 ⊕ H2, J) changes by an isomorphism. If we change the
Hamiltonians H1 and H2, the Floer complex changes by a chain homotopy. In particular, the homology
of the Floer complex does not depend on the metric, on H1, on H2, and on the orientation
data. This fact allows us to denote this graded Abelian group as HF Θ ∗ (T ∗ M). We will show in
section 4.1 that the Floer homology for figure-8 loops is isomorphic to the singular homology of
the space of figure-8 loops Θ(M),
HF Θ k (T ∗ M) ∼ = Hk(Θ(M)).
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